I am not really familiar with Sturm-Liouville theory, so possibly the answer to my question is rather trivial. Consider the SL problem \begin{equation*} L \varphi (x) : = \frac{d}{dx}\Big[ (x^2+1)^\frac 32 \frac{d\varphi}{dx} \Big] + \sqrt{1+x^2} \varphi(x) = 0. \end{equation*} In other words consider the second order linear differential equation \begin{equation} \tag{Eq 1}\label{Eq 1} (1+x^2) \varphi''(x) + 3x \varphi'(x)+\varphi(x) =0, x \in \mathbb{R}. \end{equation} One can write down the space of solutions of this equation which is the span of the functions $(1+x^2)^{-\frac 12}, (1+x^2)^{-\frac 12}\sinh^{-1}(x)$. My question is the following; does there exist a supersolution $u\in C^2(\mathbb{R})$ of \eqref{Eq 1} such that there exists $C>0$ satisfying $$ \frac 1C u(x) \leq \frac{1}{x^2+1} \leq C u(x), \forall x \in \mathbb{R}. $$ In fact, even a weaker than $C^2(\mathbb{R})$ would be enough as long as it is a supersolution in the distributional sense. I feel like the answer should be negative but I don't have a method to prove it.
Even a reference to some related results would be interesting.
